A Definition for Large Eddy Simulation Approximations of the Navier-Stokes Equations

A DEFINITION FOR LARGE EDDY SIMULATION APPROXIMATIONS OF THE NAVIER-STOKES EQUATIONS

J.-L. GUERMOND1,3 AND S. PRUDHOMME2

Abstract. This paper proposes a mathematical definition of Large Eddy Sim- ulation (LES) for three-dimensional turbulent incompressible viscous flows. We propose to call LES approximation of the Navier–Stokes equations any sequence of finite-dimensional approximations of the velocity and pressure fields that converge (possibly up to subsequences) in the Leray class to a suitable weak solution.

1. Introduction 1.1. Introductory comments. At the present time, computer predictions of turbulence phenomena by so-called Direct Numerical Simulation (DNS) of the Navier- Stokes equations are still considered a formidable task for Reynolds numbers larger than a few thousands. Since the times of Boussinesq and Reynolds, numerous turbulence models based on time-averaged or space-averaged quantities (Reynolds Averaged Navier–Stokes models, k- models, etc.) have been developed and then used in engineering applications as a means of overcoming, though often with lim- ited success, the lack of sufficient computer resources required by DNS. During the past forty years a new class of turbulence models, collectively known as Large Eddy Simulation (LES) models [23], has emerged in the literature. These models are founded on the observation that representing the whole range of flow scales may not be important in many engineering applications as one is generally inter- ested only in the large scale features of the flow. With this objective in mind, LES modelers have devised artifacts for representing the interactions between the unreachable small scales and the large ones. An extended variety of LES models is now available; see e.g. [6, 12, 18] for reviews. However no satisfactory mathematical theory for LES has yet been proposed (for preliminary attempts of formalization see [15, 19, 18]). More surprisingly, no mathematical definition of LES has been stated either (to the best of our knowledge), although some qualitative attempts have been made in this direction. For instance the following formal definition is proposed in [8]: “We define a large eddy simulation as any simulation of a turbulent flow in which the large-scale motions are explicitly resolved while the small-scale motions are represented approximately by a model (in engineering nomenclature) or parameterization (in the geosciences).” The objective of this paper is to go beyond qualitative statements like the one above by proposing a mathematical definition of LES approximations of the Navier–Stokes equations.

Date: Draft version: May 6, 2004. 1991 Mathematics Subject Classiﬁcation. 35Q30, 65N35, 76M05. Key words and phrases. Navier–Stokes equations, Turbulence, Large Eddy simulation. 1 2 J.-L. GUERMOND AND S. PRUDHOMME

This paper is organized as follows: in the remainder of the Introduction, we recall the notion of suitable weak solutions of the Navier-Stokes equations as it will be included into our definition of LES approximations. We state in Section 2 the definition that we propose for LES approximations and discuss some of its features. In particular, we introduce the concept of pre-LES-models. In Section 3, we list existing models that fall into the category of the pre-LES-models. We proceed in Section 4 by providing examples of LES approximations and for each of these examples we show how to use our definition of LES to determine the relationship between the discretization parameter and the cutoff scale parameter. Concluding remarks are finally reported in Section 5. 1.2. Suitable weak solutions. It is generally accepted that the Navier–Stokes equations stand as a reasonable model to predict the behavior of turbulent incompressible flows of viscous fluids. Upon denoting by Ω ⊂ R3 the open smooth connected domain occupied by the fluid, ]0,T [ some time interval, u the velocity field, and p the pressure, the problem is formulated as follows: 2 ∂tu + u·∇u + ∇p − ν∇ u = f in QT ,  ∇·u = 0 in QT , (1.1) u| = 0 or u is periodic,  Γ  u|t=0 = u0, where QT = Ω × (0,T ), Γ is the boundary of Ω, u0 the solenoidal initial data, f a source term, ν the viscosity, and the density is chosen equal to unity. Henceforth, we assume that (1.1) has been nondimensionalized, i.e. ν is the inverse of the Reynolds number. To implicitly account for boundary conditions, we introduce ( H1(Ω) If Dirichlet conditions, (1.2) X = 0 {v ∈ H1(Ω), v periodic} If periodic conditions. (1.3) V = {v ∈ X, ∇·v = 0}, L2 (1.4) H = V In mathematical terms, the turbulence question is an elusive one. Since the bold definition of turbulence by Leray in the 1930’s [24], calling solution turbulente any weak solution of the Navier–Stokes equations, progress has been frustratingly slow. The major obstacle in analyzing the Navier–Stokes equations has to do with the question of uniqueness of solutions in three dimensions, a question not yet solved owing to the possibility that the occurrence of so-called vorticity bursts reaching scales smaller than the Kolmogorov scale cannot be excluded. If weak solutions are not unique, a fundamental question is then to distinguish the physically relevant solutions. A possible piece of the maze might be the notion of suitable weak solutions proposed by Scheffer [30]: Definition 1.1 (Scheffer). A weak solution to the Navier-Stokes equation (u, p) 2 ∞ 2 3 is suitable if u ∈ L (0,T ; X) ∩ L (0,T ; L (Ω)), p ∈ L 2 (QT ) and the local energy balance 1 1 1 (1.5) ∂ ( u2) + ∇·(( u2 + p)u) − ν∇2( u2) + ν(∇u)2 − f · u ≤ 0 t 2 2 2 is satisfied in the distributional sense. A DEFINITION OF LES 3

It is remarkable that, to date, it is for the class of suitable weak solutions that the best partial regularity result has been proved, as stated in the so-called Caffarelli- Kohn-Nirenberg (CKN) Theorem, see Caffarelli et al. [2], Lin [25], Scheffer [30]. This theorem states that the one-dimensional Hausdorff measure of singular points of suitable solutions is zero. In other words, suitable weak solutions are almost classical (i.e., smooth). Whether these solutions are indeed classical is still far from being clear (see e.g. Scheffer [31]). Nevertheless, the result of the CKN Theorem is our principal motivation for incorporating the notion of suitable weak solutions in the definition of LES models. Note that, by analogy with nonlinear conservative laws, the condition (1.5) can be viewed as an entropy-like condition in the sense that it is expected to select physical solutions of (1.1). Duchon and Robert [7] have given an explicit form of the distribution D(u) that is missing in the left-hand side of (1.5) to reach equality. For a smooth flow, the distribution D(u) is zero; but for nonregular flow D(u) may be nontrivial. Suitable solutions are those which satisfy D(u) ≥ 0, i.e., if singularities appear, only those that dissipate energy are admissible.

2. The main definition We believe that Large Eddy Simulation approximations should be defined as solutions of finite-dimensional problems which can be implemented on digital comput- ers. In this respect, a reasonable definition should fill the gap between the current so-called LES modeling theories and their various numerical implementations. The existence of this gap has been repeatedly acknowledged in the literature without being seriously addressed; see e.g. Ferziger [9]: “In general, there is a close connection between the numerical methods and the modeling approach used in simulation; this connection has not been sufficiently appreciated by many authors.” Finally, LES approximations should select physical solutions of the Navier-Stokes equations under the appropriate limiting process. Hence we propose the following definition for LES:

∞ 2 2 Definition 2.1. A sequence (uγ , pγ )γ>0 with uγ ∈ L (0,T ; L (Ω)) ∩ L (0,T ; X) 0 2 and pγ ∈ D (]0,T [,L (Ω)) is said to be a LES approximation to (1.1) if 2 (i) There are two finite-dimensional vectors spaces Xγ ⊂ X and Mγ ⊂ L (Ω) 1 0 such that uγ ∈ C ([0,T ]; Xγ ) and pγ ∈ C ([0,T ]; Mγ ) for all T > 0. (ii) The sequence converges (up to subsequences) to a weak solution of (1.1), 2 0 2 say uγ * u weakly in L (0,T ; X) and pγ → p in D (]0,T [,L (Ω)). (iii) The weak solution (u, p) is suitable. At this point, we want to emphasize that two parameters are actually hidden in the above definition. Since Xγ and Mγ are finite-dimensional, there is a discretization parameter h associated with the size of the smallest scale that can be 3 represented in Xγ , roughly dim(Xγ ) = O((L/h) ) where L = diam(Ω). In the following we also consider a regularization parameter ε associated with some regularization of the Navier–Stokes equations. This parameter is the lengthscale of the smallest eddies that are allowed to be nonlinearly active in the flow (large eddies). In the above definition, the parameter γ is yet an unspecified combination of the two parameters h and ε. In practice, the construction of a LES model will be decomposed into the following three steps: 4 J.-L. GUERMOND AND S. PRUDHOMME

(1) Construction of what we hereafter call the pre-LES-model. This step consists of regularizing the Navier-Stokes equations by introducing the parameter ε. The purpose of the regularization technique is to yield a well-posed problem for all times. Moreover, the limit solution of the pre-LES-model must be a weak solution to the Navier–Stokes equations as ε → 0 and should be suitable. (2) Discretization of the pre-LES-model. This step introduces the meshsize parameter h and the finite-dimensional spaces Xγ , Mγ . (3) Determination of a (possibly maximal) relationship between ε and h. The parameters ε and h should be selected in such a way that the discrete solution is ensured to converge to a suitable solution of the Navier-Stokes equations when ε → 0 and h → 0. The novelty in the proposed definition is that enforcing the limit solution (u, p) to be suitable yields a constraint on the limiting processes limε→0 and limh→0. Because of this constraint, we prefer to use the neutral parameter γ than either one of the two, and whenever we write γ → 0, it will be understood that ε → 0 and h → 0 in some manner yet to be specified. A first question that comes to mind is the following: Is a sequence of Direct Numerical Solutions (DNS) a LES approximation as defined above? To clarify this issue let us consider the construction proposed by Hopf [17]. Let Xh ⊂ X and 2 Mh ⊂ L (Ω) be two finite-dimensional vector spaces and consider the following Galerkin approximation

 (∂tuh, v) + b(uh, uh, v) − (ph, ∇·v) + ν(∇uh, ∇v)   = (f, v) ∀t ∈ [0,T ], ∀v ∈ Xh, (2.1) (q, ∇·u ) = 0, ∀t ∈ [0,T ], ∀q ∈ M ,  h h  (uh|t=0, v) = (u0, v), ∀v ∈ Xh, where (·, ·) stands for the L2-scalar product, b is a trilinear form accounting for the nonlinear advection term and is skew-symmetric with respect to its second 1 and third arguments. For instance b(uh, uh, v) = (uh·∇uh, v) − 2 (uh∇·uh, v) and b(uh, uh, v) = ((∇×uh)×uh, v) are admissible candidates. The above construction is usually referred to as DNS in the Computational Fluid Dynamics literature. Owing to standard a priori estimates uniform in h, it is clear that the approximate solution complies with items (i) and (ii) of the above definition; see, e.g., Lions [26, 27] or Temam [34]. However, it is not yet known whether such a construction yields a suitable solution at the limit when h → 0, i.e., surprisingly item (iii) is not guaranteed to hold. In other words, it may happen that the solution sets spanned by the limits of DNS approximations and LES approximations are not identical. The answer to this question seems to be intimately linked with the regularity question, which, we recall, is the object of one the seven prizes offered by the Clay institute. Definition 2.1 is based on the conjecture that the LES solution set is strictly smaller than the DNS solution set in general.

3. Review of existing pre-LES-models We now briefly survey regularization techniques that we will consider as pre- LES-models. These models do not comply with item (i) of Definition 2.1 since both A DEFINITION OF LES 5 uε and pε have values in infinite-dimensional vector spaces, but they all satisfy items (ii) and (iii). 3.1. Hyperviscosity. Lions [26, 27] proposed the following hyperviscosity perturbation of the Navier–Stokes equations:  2 2α 2 α ∂tuε + uε·∇uε + ∇pε − ν∇ uε + ε (−∇ ) uε = f in QT ,  ∇·uε = 0 in QT , (3.1) u | , ∂ u | , . . . , ∂α−1u | = 0, or u is periodic  ε Γ n ε Γ n ε Γ ε  u|t=0 = u0, where ε > 0 and α is an integer. The appealing aspect of this perturbation is 5 that it yields a wellposed problem in the classical sense when α ≥ 4 in three space dimensions. More precisely, upon denoting by d ≥ 2 the space dimension, the following result (see [16, 26, 27]) holds 2 2 α Theorem 3.1. Assume f ∈ L (0,T ; L (Ω)) and u0 ∈ H (Ω) ∩ X. Problem (3.5) ∞ α d+2 has a unique solution uε in L (0,T ; H (Ω)∩X) for all times T > 0 if α ≥ 4 . Up 2 to subsequences, uε converges to a weak solution u of (1.1), weakly in L (0,T ; X). Moreover, if periodic boundary conditions are enforced, the limit solution (u, p) is suitable. It is remarkable that hyperviscosity models are frequently used in computer simulations for oceanic and atmospheric flows [1]. 3.2. Leray mollification. A simple construction yielding a suitable solution has indeed been proposed by Leray [24] before this very notion was introduced in the literature. Leray proved the existence of weak solutions by using a, now very popular, mollification technique. Assume that Ω is the three-dimensional torus (0, 2π)3. Denoting by B(0, ε) ⊂ R3 the ball of radius ε centered at 0, consider a sequence of mollifying functions (φε)ε>0 satisfying: ∞ 3 R (3.2) φε ∈ C ( ), supp(φε) ⊂ B(0, ε), 3 φε(x) dx = 1. 0 R R R Defining the convolution product φε∗v(x) = 3 v(y)φε(x − y) dy, Leray suggested R to regularize the Navier–Stokes equations as follows:  2 ∂tuε + (φε ∗uε)·∇uε + ∇pε − ν∇ uε = φε ∗f,   ∇·u = 0, (3.3) ε  uε is periodic,   uε|t=0 = φε ∗u0. The following holds (see [24] and [7]): ∞ Theorem 3.2. For all u0 ∈ H, f ∈ H, and ε > 0, (3.3) has a unique C solution. The velocity is bounded uniformly in L∞(0,T ; H) ∩ L2(0,T ; V) and one subsequence converges weakly in L2(0,T ; V). The limit solution as ε → 0 is a suitable weak solution of the Navier–Stokes equations. Hence the above construction complies with items (ii) and (iii) of Definition 2.1. The mollification technique can be extended to account for the homogeneous Dirich- let boundary condition as done in [2]. The limit solution is suitable in this case as well. 6 J.-L. GUERMOND AND S. PRUDHOMME

Remark 3.1. Roughly speaking, the convolution process removes scales that are smaller than ε. Hence, by using φε∗uε as the advection velocity, scales smaller than ε are not allowed to be nonlinearly active. 3.3. NS-α and Leray-α models. Introduce the so-called Helmholtz ﬁlter (·): v 7→ v such that (3.4) v := (I − ε2∇2)−1v, where either homogeneous Dirichlet boundary conditions or periodic boundary conditions are enforced depending on the setting considered. The so-called Navier– Stokes–alpha model introduced in Chen et al. [4] and Foias et al. [10, 11] consists of the following:  T 2 ∂tuε + uε·∇uε + (∇uε) · uε − ν∇ uε + ∇πε = f,   ∇·u = 0, (3.5) ε  uε|Γ = 0, uε|Γ = 0, or uε, and uε are periodic,   uε|t=0 = u0, Once again, regularization yields uniqueness as stated in the following

Theorem 3.3 (Foias, Holm and Titi [10, 11]). Assume f ∈ H, u0 ∈ V. Problem 0 (3.5) with the Helmholtz filter (3.4) has a unique solution uε in C ([0,T ); V) with 2 ∞ ∂tuε ∈ L (]0,T [; H). The solution uε is uniformly bounded in L (0, +∞; H) ∩ 2 2 L (0, +∞; V) and one subsequence converges weakly in Lloc(0, +∞; V) to a weak Navier–Stokes solution as ε → 0. Here again, it is a simple matter to show that when periodic boundary conditions are enforced uε converges, up to subsequences, to a suitable solution. A variant of the above regularization technique consists of replacing the term T 1 2 (∇uε) · uε in (3.5) by ∇ 2 uε. The resulting model then falls in the class of the Leray regularization in the sense that the momentum equation is the same as that in (3.3) but for the advection velocity φε ∗uε which is replaced by uε. This model has been analyzed in [5] and is called the Leray-α model; see also [13]. 3.4. Nonlinear viscosity models. Recalling that the Navier–Stokes equations are based on Newton’s linear hypothesis, Ladyˇzenskaja and Kaniel proposed to modify the incompressible Navier–Stokes equations to take into account possible large velocity gradients, [20, 21, 22]. Ladyˇzenskaja introduced a nonlinear viscous tensor Tij(∇u), 1 ≤ i, j ≤ 3 satisfying the following conditions: 1 L1. T is continuous and there exists µ ≥ 4 such that 3×3 2µ (3.6) ∀ ξ ∈ R , |T(ξ)| ≤ c(1 + |ξ| )|ξ|. L2. T satisfies the coercivity property: 3×3 2 0 2µ (3.7) ∀ ξ ∈ R , T(ξ): ξ ≥ c|ξ| (1 + c |ξ| ). L3. T possesses the following monotonicity property: There exists a constant c > 0 such that for all solenoidal fields ξ, η in W1,2+2µ(Ω) either coinciding on the boundary Γ or being periodic, Z Z (3.8) (T(∇ξ) − T (∇η)) : (∇ξ − ∇η) ≥ c |∇ξ − ∇η|2. Ω Ω A DEFINITION OF LES 7

The three above conditions are satisﬁed if (3.9) T(ξ) = β(|ξ|2)ξ, provided the viscosity function β(τ) is a positive monotonically-increasing function of τ ≥ 0 and for large values of τ the following inequality holds cτ µ ≤ β(τ) ≤ c0τ µ, 1 0 with µ ≥ 4 and c, c are some strictly positive constants. Introducing a positive constant ε > 0, the modiﬁed Navier–Stokes equations take the form 2µ+1  ∂tuε + uε·∇uε + ∇pε − ∇·(ν∇uε + ε T(∇uε)) = f,   ∇·uε = 0 (3.10) u | = 0, or u is periodic,  ε Γ ε  uε|t=0 = u0. The main result from [22, 21] (see [20] for a similar result where monotonicity is also assumed) is the following theorem

2 2 Theorem 3.4. Assume u0 ∈ H and f ∈ L (]0, +∞[; L (Ω)). Provided conditions L1, L2, and L3 hold, then (3.10) has a unique weak solution, for all R > 0, in L2+2µ(]0,T [; W1,2+2µ(Ω) ∩ V) ∩ C0([0,T ]; H). This result states that a small appropriate amount of nonlinear viscosity is actually suﬃcient to ascertain that the energy cascade stops, which automatically translates into uniqueness of the solution for arbitrary times. Moreover, for periodic boundary conditions, (uε, pε) converges, up to subsequences, to a suitable solution of (1.1) as ε → 0. Possibly one of the most popular models for Large Eddy Simulation is that proposed by Smagorinsky [32], which corresponds to setting T(∇u) = |D|D. For µ 1 this model β(τ) = τ with µ = 2 . In conclusion, perturbing the Navier–Stokes equations with a term like Smagorin- sky’s model solves the uniqueness question and ensures that the limit solution are suitable. Note however that, at this point, item (i) of Deﬁnition 2.1 does not hold; i.e., the above arguments do not give guidelines on the way ε should be linked to a discretization parameter like the meshsize h.

4. Discretization The purpose of this section is to illustrate Deﬁnition 2.1. We introduce discrete versions of some of the pre-LES-models described above. In each case we determine the relationship between the parameters h and ε so that the resulting approximate solutions converge to suitable solutions when the parameters go to zero. For the sake of simplicity, we restrict ourselves to periodic boundary conditions and spectral approximation techniques.

4.1. The discrete hyperviscosity model. We turn our attention to the hyperviscosity model introduced in Section 3.1. We construct a Galerkin-Fourier approximation and we show how Deﬁnition 2.1 enforces strong constraints on the relation linking the hyperviscosity coeﬃcient ε to the polynomial degree of the approximation. 8 J.-L. GUERMOND AND S. PRUDHOMME

` For any z ∈ C , 1 ≤ ` ≤ 3, we denote by |z| the Euclidean norm and by |z|∞ the maximum norm. We denote by z the conjugate of z. Recall that Sobolev spaces Hs(Ω), s ≥ 0, can be equivalently defined in terms of Fourier series as follows s P ik·x P 2 s 2 H (Ω) = u = 3 uke , uk = u−k, 3 (1 + |k| ) |uk| < +∞ . k∈Z k∈Z In other words, the set of trigonometric polynomials exp(ik ·x), k ∈ Z3, is complete and orthogonal in Hs(Ω) for all s ≥ 0. The scalar product in L2(Ω) is denoted by −3 R s −s (u, v) = (2π) Ω uv and the dual of H (Ω) by H (Ω). We introduce the closed subspace H˙ s(Ω) of Hs(Ω) composed of the functions of zero mean value. Let N be a positive integer, henceforth referred to as the cutoff wave number. We introduce the set of trigonometric polynomials of partial degree less than or equal to N: n o = p(x) = P c eik·x, c = c , PN |k|∞≤N k k −k ˙ and denote by PN the subspace of PN composed of the trigonometric polynomials of zero mean value. Finally, to approximate the velocity and the pressure fields we introduce the following finite-dimensional vector spaces: ˙ 3 ˙ (4.1) XN = PN , and MN = PN . The construction of the hyperviscosity model involves the definition of a large eddy scale εN and a hyperviscosity kernel Q(x). We choose to build the kernel in such a way that it acts only on the high wave numbers of the velocity field, namely Ni ≤ |k|∞ ≤ N, Ni standing for an intermediate cutoff wave number yet to be fixed. This idea is similar to the spectral viscosity technique that Tadmor [33, 28, 3] developed for nonlinear scalar conservation laws. We start by introducing the hyperviscosity parameter α such that 5 (4.2) α > . 4 We then introduce θ, with 0 < θ < 1, from which we define the large scale parameter Ni and the corresponding large eddy cutoff scale εN

θ 1 (4.3) Ni = N , εN = Ni We also introduce a hyperviscosity kernel Q(x) as follows: X (4.4) Q(x) = (2π)−3 |k|2αeik·x

Ni≤|k|∞≤N

The kernel Q is such that for all vN ∈ XN Z X 2α ik·x (4.5) Q∗vN (x) = vN (y)Q(x − y)dy = |k| vke Ω Ni≤|k|∞≤N When α is an integer, Q∗(·) is the α-th power of the restriction of the Laplace operator on the space spanned by the Fourier modes comprised between Ni and N. Note that when θ increases, the vanishing viscosity amplitude decreases and the range of wavenumbers on which the kernel Q(x) is active shrinks. A DEFINITION OF LES 9

The spectral hyperviscosity model consists of the following: 1 0 Seek uN ∈ C ([0,T ]; XN ) and pN ∈ C ([0,T ]; MN ) such that   (∂ u , v) + (u ·∇u , v) − (p , ∇·v) + ν(∇u , ∇v)  t N N N N N  2α (4.6) + εN (Q∗uN , v) = (f, v), ∀v ∈ XN , ∀t ∈ (0,T ],  (∇·u , q) = 0, ∀q ∈ M , ∀t ∈ (0,T ],  N N  (uN , v)|t=0 = (u0, v), ∀v ∈ XN ,. One interesting feature of the above technique is the following Proposition 4.1. The hyperviscosity perturbation is spectrally small, i.e., 2α −θs s (4.7) εN kQ∗vN kL2 . N kvN kHs , ∀vN ∈ H (Ω), ∀s ≥ 2α. The interpretation of the above result is that the consistency error induced by the hyperviscosity is arbitrarily small if the exact solution of the Navier–Stokes equation is smooth. The main result of this section is the following: 2 2 α Theorem 4.1. Let f ∈ L (0,T ; L (Ω)) and u0 ∈ H (Ω) ∩ V. Assume that (4.2) and (4.3) hold. Moreover, assume that ( 4α−5 3 4α If α ≤ 2 , (4.8) 0 < θ < 2(α−1) 2α+3 Otherwise. 2 then, up to subsequences, the solution uN to (4.6) converges weakly in L (0,T ; V) p q 4q and strongly in any L (0,T ; L (Ω)), with 2 ≤ p < 3(q−2) < +∞, to a suitable solution of (1.1) as N goes to infinity. The above result, proved in [16], can be interpreted as follows. Since the hyperviscosity term is meant to be a perturbation of the Navier–Stokes equations, one would want this term to be as small as possible. In fact, one would like θ to be as large as possible; the choice θ = 1 is that which minimizes the impact of the hyperviscosity perturbation. But, if θ is too close to one, the hyperviscosity term cannot play the role it is assigned, i.e., the limit solution cannot be guaranteed to be suitable (see item (iii)). It is shown in [16] that a sufficient condition for the limit solution to be suitable is that the bound from above in (4.8) holds. In this sense we claim that Definition 2.1 is constructive. Numerically speaking, the above result is somewhat vague. A possible algorithm 4α−5 for implementing this hyperviscosity model is to pick α, to take θ < 4α or θ < 2(α−1) θ −1 2α+3 , depending on the value of α, and to finally set Ni = c1N and εN = c2Ni . In this case, α is a free parameter and the constants c1, c2 are coefficients that can be played with, provided they are of order one. Some admissible values of the parameters α and θ are shown in Table 1.

3 α 2 2 3 4 5 1 2 4 6 8 θ < 6 < 7 < 9 < 11 < 13

Table 1. Admissible values of the parameters α and θ. 10 J.-L. GUERMOND AND S. PRUDHOMME

Remark 4.1. With the polynomial degree N we can associate the mesh size hN = −1 N . Then we observe that the condition (4.8), i.e. θ < 1, means that hN εN . In other words, the scales filtered by the hyperviscosity are significantly larger than the grid size. 4.2. The discrete Leray model. For the sake of simplicity, we restrict the analysis to the periodic case, i.e. Ω is assumed to be the three-dimensional torus, and we reuse the Fourier setting introduced in Section 4.1. Let N be an integer and set ˙ 3 ˙ (4.9) XN = PN , and MN = PN . Introduce θ such that 0 < θ < 1 and set θ (4.10) Ni = N . We also consider the truncation operator P : Hs(Ω) −→ 3 such that Ni PNi X X P : Hs(Ω) 3 v eik·x = v 7−→ v eik·x ∈ 3 Ni k k PNi 3 k∈Z |k|∞≤Ni and let QNi denote the operator QNi = I − PNi . In the Fourier setting, a simple way of regularizing the nonlinear term consists of removing the high frequencies from the advection velocity. Thus, instead of using uN as advection field, we use

PNi uN . The discrete Leray model then reads: Seek u ∈ C1([0,T ]; X ) and p ∈ C0([0,T ]; M ) such that  N N N N  for all t ∈ (0,T ], for all v ∈ XN , and for all q ∈ MN ,  (4.11) (∂tuN , v) + (PNi uN ·∇uN , v) − (pN , ∇·v) + ν(∇uN , ∇v) = (f, v),  (∇·uN , q) = 0,  (uN , v)|t=0 = (u0, v).

2 2 Theorem 4.2. Let f ∈ L (0,T ; L (Ω)) and u0 ∈ H. Assume that (4.10) hold. If 2 (4.12) θ < , 3 2 the solution uN to (4.11) converges weakly, up to subsequences, in L (0,T ; V) and p q 4q strongly in any L (0,T ; L (Ω)), with 2 ≤ p < 3(q−2) < +∞, to a suitable solution of (1.1) as N goes to inﬁnity. Proof. We just sketch the proof since the details are similar to those given in the proof of Theorem 5.1 in [16]. The main diﬃculty revolves around the handling of the nonlinear term when proving that the limit solution is suitable. As usual, the basic a priori estimates are Z T 2 2 2 (4.13) kuN kL2 + k∇PNi uN kL2 + k∇QNi uN kL2 ≤ c. 0

Let φ ∈ D(QT ) and use PN (φuN ) to test the momentum equation in (4.11). The nonlinear term gives

(PNi uN ·∇uN ,PN (uN φ)) = (PNi uN ·∇uN , uN φ) + R1 1 2 = −( 2 |uN | PNi uN , ∇φ) + R1, 1 2 = −( 2 |uN | uN , ∇φ) + R1 + R2, A DEFINITION OF LES 11 where

R1 = (PNi uN ·∇uN ,PN (uN φ) − uN φ), 1 2 R2 = ( 2 |uN | QNi uN , ∇φ). The ﬁrst residual is handled as follows:

∞ 2 2 |R1| ≤ kPNi uN kL k∇uN kL kPN (uN φ) − uN φkL , 3 −1 2 2 1 1 . Ni N kPNi uN kL kuN kH kuN φkH , 3 2 θ−1 2 . N kuN kL2 kuN kH1 kφkW 1,∞ , where . means that the inequality holds up to a constant independent of N. Then, R T it is clear that 0 |R1| → 0 as N → ∞ owing to (4.12). In a similar fashion, we R T have that 0 |R2| → 0 as N → ∞. The rest of the proof follows as that of Theorem 5.1 in [16]. −1 Remark 4.2. Denoting by εN = Ni the regularization scale (the large eddy scale) 3 −1 2 and hN = N the discretization scale, Theorem 4.2 shows that if εN hN , then the pair (uN , pN ) is a LES approximation in the sense of Deﬁnition 2.1. 4.3. The discrete Leray-α model. Still keeping the Fourier framework, we now consider the following discrete version of the Leray-α model introduced at the end of Section 3.3; see also [5, 14]:  1 0 Seek uN ∈ C ([0,T ]; XN ) and pN ∈ C ([0,T ]; MN ) such that   for all t ∈ (0,T ], for all v ∈ XN , and for all q ∈ MN ,  (∂tuN , v) + (u¯N ·∇uN , v) − (pN , ∇·v) + ν(∇uN , ∇v) = (f, v), (4.14) 2 (u¯N , v) + εN (∇u¯N , ∇v) = (uN , v),   (∇·uN , q) = 0,  (uN , v)|t=0 = (u0, v), where εN is the scale of the smallest eddies that we authorize to be nonlinearly active: −1 −θ (4.15) εN = Ni = N , 0 < θ < 1. Note that the system (4.14) is similar to (4.11) except for the advective velocity

PNi uN which is now replaced by the regularized velocity u¯N . 2 2 Theorem 4.3. Let f ∈ L (0,T ; L (Ω)) and u0 ∈ H. Assume that (4.15) holds. If 2 (4.16) θ < , 3 2 the solution uN to (4.14) converges weakly, up to subsequences, in L (0,T ; V) and p q 4q strongly in any L (0,T ; L (Ω)), with 2 ≤ p < 3(q−2) < +∞, to a suitable solution of (1.1) as N goes to inﬁnity.

Proof. Using the fact that ∇·u¯N = 0 and taking v = PN (uN φ), the nonlinear term becomes 1 2 (u¯N ·∇uN ,PN (uN φ)) = (u¯N ·∇uN , uN φ) + R = −( 2 |uN | u¯N , ∇φ) + R, 12 J.-L. GUERMOND AND S. PRUDHOMME with

|R| = |(u¯N ·∇uN ,PN (uN φ) − uN φ)|

. ku¯N kL∞ k∇uN kL2 kPN (uN φ) − uN φkL2 −1 2 . N ku¯N kL∞ kuN kH1 kφkW 1,∞ . 2 Using ku¯N kL∞ . k∇u¯N kL2 k∆u¯N kL2 and the bounds εN k∇u¯N kL2 . kuN kL2 and 2 εN k∆u¯N kL2 . kuN kL2 (from the Helmholtz problem), an estimate of the residual is 3 − 2 −1 2 |R| . εN N kuN kL2 kuN kH1 kφkW 1,∞ 3 2 θ−1 2 . N kuN kL2 kuN kH1 kφkW 1,∞ . R T 2 It follows that 0 |R1| → 0 as N → ∞ provided that θ < 3 . In other words, the discrete solution (uN , pN ) converges to a suitable weak solution of the Navier-Stokes 3 2 equations whenever εN hN . 4.4. The other discrete models. We have not yet been able to show that discrete counterparts of the other pre-LES-models introduced in Section 3, namely the NS-α model and the nonlinear viscosity models, satisfy the LES definition. Regarding the nonlinear viscosity models, we are still facing technical issues to prove that the discrete solutions actually converge to suitable weak solutions of the Navier-Stokes equations. In particular, the difficulty lies in the fact that the Fourier analysis is not the proper tool to work with Lp spaces when p 6= 2. However, we believe that these issues, being purely technical, will be eventually overcome in the near future. T In the case of the NS-α model, the presence of the term (∇uε) · uε poses some difficulties when passing to the limit. Whether these difficulties are either technical or fundamental is not yet clear. Surprisingly we have also observed that the Nonlinear Galerkin Method [29] can be reinterpreted in the light of Definition 2.1 as a LES technique. Indeed, following a similar approach as above, it can be shown that the discrete solution of a slightly modified version of the Nonlinear Galerkin Method yields a suitable solution in the limit under appropriate conditions, (the modified version in question consists of replacing ∂tuN by ∂tPNi uN in (4.11)). This interpretation of the Nonlinear Galerkin Method will be the subject of a forthcoming paper.

5. Conclusions A definition for LES approximations of the Navier-Stokes equations has been proposed in this paper. The proposed definition introduces two parameters: a discretization scale h and a cutoff scale for the large eddies ε. The ratio h/ε should be chosen so that the limit solution is suitable. It is proven on three examples that h should be much smaller than ε in order to ensure that the approximate solutions converge to a suitable solution of the Navier-Stokes equations; see Remarks 4.1 and 4.2. This result sheds some doubts on what is often suggested in the literature, namely that ε should be of the same order as h. Whether the mathematical framework proposed herein really models turbulence is far from being clear, and we do not make any claim in this respect. We identify two obstacles in the way. First, although the notion of turbulence is quite intuitive A DEFINITION OF LES 13 and is a daily experience, the very concept of turbulence has yet to be mathemati- cally defined. In particular, we are not aware of any other mathematical definition of turbulence than that proposed by Leray, who identifies “solutions turbulentes” and weak solutions. Second, there is still a possibility that Galerkin-based solutions could eventually be shown to be suitable. In this event, the framework we proposed for LES would be either irrelevant or should be adapted to the inviscid Euler equations. This last argument shows again that the question of the regularity of the Navier–Stokes equations is far from being of academic interest only.

References [1] C. Basdevant, B. Legras, R. Sadourny, and M. Béland. A study of barotropic model flows: intermittency, waves and predictability. J. Atmos. Sci., 38:2305–2326, 1981. [2] L. Caffarelli, R. Kohn, and L. Nirenberg. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math., 35(6):771–831, 1982. [3] G.-Q. Chen, Q. Du, and E. Tadmor. Spectral viscosity approximations to multidimensionnal scalar conservation laws. Math. Comp., 61:629–643, 1993. [4] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi, and S. Wynne. A connection between the Camassa-Holm equation and turbulent flows in channels and pipes. Physics of Fluids, 11(8):2343–2353, 1999. [5] A. Cheskidov, D. D. Holm, E. Olson, and E. S. Titi. On a Leray-α model of turbulence. Royal Society London, Proceedings, Series A, Mathematical, Physical & Engineering Sci- ences, 2004. Submitted. [6] J.P. Chollet, P. R. Voke, and L. Kleiser, editors. Direct and large-eddy simulation. II, volume 5 of ERCOFTAC Series. Kluwer Academic Publishers, Dordrecht, 1997. [7] J. Duchon and R. Robert. Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity, 13(1):249–255, 2000. [8] J.H. Ferziger. Subgrid-scale modeling. In Boris Galperin and Steven A Orszag, editors, Large Eddy Simulation of Complex Engineering and Geophysical Flows, CRM Proceedings & Lec- ture Notes. Cambridge University Press, 1993. [9] J.H. Ferziger. Direct and large eddy simulation of turbulence. In A. Vincent, editor, Numerical Methods in Fluid mechanics, volume 16 of CRM Proceedings & Lecture Notes. Centre de Recherches Mathématiques, Universitéde Montréal, American Mathematical Society, 1998. [10] C. Foias, D.D. Holm, and E.S. Titi. The Navier–Stokes-alpha model of fluid turbulence. Physica D, 152-153:505–519, 2001. [11] C. Foias, D.D. Holm, and E.S. Titi. The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory. J. Dynam. Differential Equations, 14(1):1–35, 2002. [12] B. J. Geurts. Elements of Direct and Large-Eddy Simulation. Springer-Verlag, R.T. Edwards, Inc., Philadelphia, PA, 2004. [13] B. J. Geurts and D. D. Holm. Regularization modeling for large-eddy simulation. Phys. Fluids, 15(1):L13–L16, 2003. [14] J.-L. Guermond, J.T. Oden, and S. Prudhomme. On a relation between the leray regularization and the Navier–Stokes alpha-model. Physica D, 177(1–4):23–30, 2003. [15] J.-.L. Guermond, J.T. Oden, and S. Prudhomme. Some mathematical issues concerning large eddy simulation models of turbulent flows. J. Math. Fluid Mech., 6(2):194–248, 2004. [16] J.-L. Guermond and S. Prudhomme. Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows. M2AN Math. Model. Numer. Anal., 37(6):893– 908, 2003. [17] E. Hopf. Uber¨ die Anfangswertaufgabe fürdie hydrodynamischen Grundgleichungen. Math. Nachr., 4:213–231, 1951. [18] V. John. Large eddy simulation of turbulent incompressible flows, volume 34 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin, 2004. [19] V. John and W. J. Layton. Analysis of numerical errors in large eddy simulation. SIAM J. Numer. Anal., 40(3):995–1020 (electronic), 2002. [20] S. Kaniel. On the initial value problem for an incompressible fluid with nonlinear viscosity. J. Math. Mech., 19(8):681–706, 1970. 14 J.-L. GUERMOND AND S. PRUDHOMME

[21] O.A. Ladyˇzenskaja. Modification of the Navier–Stokes equations for large velocity gradients. In O.A. Ladyˇzenskaja, editor, Seminars in Mathematics V.A. Stheklov Mathematical Insti- tute, volume 7, Boundary value problems of mathematical physics and related aspects of function theory, Part II, New York, NY, 1970. Consultant Bureau. [22] O.A. Ladyˇzenskaja. New equations for the description of motion of viscous incompressible fluds and solvability in the large of boundary value problems for them. In O.A. Ladyˇzenskaja, editor, Proc. of the Stheklov Institute of Mathematics, number 102 (1967), Boundary value problems of mathematical physics. V, Providence, RI, 1970. American Mathematical Society (AMS). [23] A. Leonard. Energy cascade in Large-Eddy simulations of turbulent fluid flows. Adv. Geo- phys., 18:237–248, 1974. [24] J. Leray. Essai sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math., 63:193– 248, 1934. [25] F. Lin. A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math., 51(3):241–257, 1998. [26] J.-L. Lions. Sur certaines équations paraboliques non linéaires. Bull. Soc. Math. France, 93:155–175, 1965. [27] J.-L. Lions. Quelques méthodes de résolution des problèmesaux limites non linéaires, volume 1. Dunod, Paris, France, 1969. [28] Y. Maday, M. Ould Kaber, and E. Tadmor. Legendre pseudospectral viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal., 30(2):321–342, 1993. [29] M. Marion and R. Temam. Nonlinear Galerkin methods. SIAM J. Numer. Anal., 26(5):1139– 1157, 1989. [30] V. Scheffer. Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys., 55(2):97–112, 1977. [31] V. Scheffer. Nearly one-dimensional singularities of solutions to the Navier-Stokes inequality. Comm. Math. Phys., 110(4):525–551, 1987. [32] J. Smagorinsky. General circulation experiments with the primitive equations, part i: the basic experiment. Monthly Wea. Rev., 91:99–152, 1963. [33] E. Tadmor. Convergence of spectral methods for nonlinear conservation laws. SIAM J. Nu- mer. Anal., 26(1):30–44, 1989. [34] R. Temam. Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland, Am- sterdam, 1984.

1LIMSI (CNRS-UPR 3152), BP 133, 91403, Orsay, France

2ICES, The University of Texas at Austin, TX 78712, USA

3Department of Mathematics, Texas A&M University, College Station, TX 77843- 3368, USA E-mail address: [email protected], [email protected] E-mail address: [email protected]